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5.80 Small-Molecule Spectroscopy and Dynamics
Fall 2008
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Lecture # 33 Supplement
Based on a lecture written by Professor Patrick H. Vaccaro.
Outline
(i) “true” Eigenstates: A long, hard climb;
(ii) the “total” molecular Hamiltonian and its Schrödinger Equation;
(iii) the electronic Schrödinger Equation;
(iv) transformation of the molecular Schrödinger Equation;
(v) the Adiabatic Approximation;
(vi) Adiabatic corrections;
(vii) Non-Adiabatic corrections;
e 1 A2 ← X
e 1 A1 absorption in H2 CO: a vibronic coupling model.
(viii) the transition moment of the A
Image removed due to copyright restrictions.
Figure 1: Various routes to approach the exact non-adiabatic wavefunction. From “What Does the Term
‘Vibronic Coupling’ Mean” by T. Azumi and K. Matsuzaki, Photochemistry and Photobiology 25, 315-326
(1977).
1
5.80 Lecture # 33 Supplement
Time-Independent Schrödinger Equation for a Molecular System
Htotal (r, Q)Ψt (r, Q) = Et Ψt (r, Q)
where
Htotal (r, Q) = Te (r) + TN (Q) + U (r, Q) + V (Q)
“r” represents electronic coordinates
“Q” represents mass-weighted nuclear coordinates describing displacements from a reference configuration
“Q0 ”
Te (r) ≈
TN (Q) ≈
−ℏ2 X ∂ 2
2me i ∂ri2
represents the electronic kinetic energy
−ℏ2 X ∂ 2
2 n ∂Q2n
represents the nuclear kinetic energy
U (r, Q) represents the Coulombic potential energy
V (Q) represents the potential energy of the nuclei
PROBLEM: Hamiltonian does not permit separation of variables. Therefore, exact solution is not possible.
Consider only the terms depending on the electronic coordinates (i.e. the so-called Electronic Hamilto­
nian)
Helec (r, Q) = Te (r) + U (r, Q)
= Te (r) + U (R, Q0 ) + ΔU (r, Q)
= Helec (r, Q0 ) + ΔU (r, Q)
where
U (r, Q) = U (r, Q0 ) + ΔU (r, Q)
Helec (r, Q0 ) = Te (r) + U (r, Q0 ).
Note that:
U (r, Q) = U (r, Q0 ) +
X ∂U (r, Q) n
∂Qn
Qn
0
1 X ∂ 2 U (r, Q)
+
Qn Qm + . . .
2 nm ∂Qn ∂Qm 0
Page 3
5.80 Lecture # 3 3 Supplement
Consequently:
ΔU (r, Q) ≈
X ∂U (r, Q) n
∂Qn
Qn +
0
1 X ∂ 2 U (r, Q)
Qn Qm + . . .
2 n,m ∂Qn ∂Qm 0
Define two types of Electronic Schrödinger Equations
(i) The Dynamical equation for Helec (r, Q)
{the “Born” representation}
Helec (r, Q)ψi (r, Q) = ǫi (Q)ψi (r, Q)
[Te (r) + U (r, Q)]ψi (r, Q) = ǫi (Q)ψi (r, Q)
dynamical electronic wavefunctions: ψi (r, Q) ⇒ Born Space
(ii) The static equation for Helec (r, Q0 )
{the “Longuet-Higgins” representation}
Helec (r, Q0 )ψi0 (r, Q0 ) = ǫ0i (Q0 )ψi0 (r, Q0 )
[Te (r) + U (r, Q0 )]ψi0 (r, Q0 ) = ǫ0i (Q0 )ψi0 (r, Q0 )
static electronic wavefunctions: ψi0 (r, Q0 ) ⇒ Longuet-Higgins Space
The Eigenstates of the Total Hamiltonian can now be expanded in either of these two electronic basis
sets:
(i) The dynamical or Born Representation:
Ψt (r, Q) =
X
ψk (r, Q)χD
kt (Q)
k
(ii) The static or Longuet-Higgins Representation:
X
Ψt (r, Q) =
ψk0 (r, Q0 )χSkt (Q).
k
Note that
ψk (r, Q) =
X
ψℓ0 (r, Q0 )λℓk (Q)
ℓ
and
Ψt (r, Q) =
X
ψk (r, Q)χD
kt (Q)
k
=
"
X X
k
=
X
ℓ
#
ψℓ0 (r, Q0 )λℓk (Q) χD
kt (Q)
ℓ
ψℓ0 (r, Q0 )χSℓt (Q).
Page 4
5.80 Lecture # 3 3 Supplement
where
χSℓt (Q) =
X
λℓk (Q)χD
kt (Q)
k
Now recall the Schrödinger Equation for the total Hamiltonian
(i) the dynamical or Born Representation:
Htotal (r, Q)Ψt (r, Q) = Et Ψt (r, Q)
[Te (r) + U (r, Q) + TN (Q) + V (Q) − Et ]Ψt (r, Q) = 0
X
[Helec (r, Q) + TN (Q) + V (Q) − Et ]
ψk (r, Q)χD
kt (Q) = 0
k
substitute the dynamical electronic Schrödinger Equation and the explicit expression for TN (Q):
"
X
{ǫk (Q) + V (Q) − Et } ψk (r, Q)χD
kt (Q)
k
#
∂ 2 χD
∂ψk (r, Q) ∂χD
ℏ X ∂ 2 ψk (r, Q) D
kt (Q)
kt (Q)
=0
χkt (Q) + ψk (r, Q)
+2
−
2 n
∂Q2n
∂Q2n
∂Qn
∂Qn
2
Multiply from the left by ψj⋆ (r, Q) and integrate over the electronic coordinates realizing that:
hψj (r, Q)|ψk (r, Q)i = δjk .
One thus obtains a set of coupled differential equations:
{TN (Q) + V (Q) + ǫj (Q) + hψj (r, Q)|TN (Q)|ψj (r, Q)i − Et } χD
jt (Q)
)
X
X
∂ ∂
2
ψk (r, Q)
+
ψj (r, Q) χD
hψj (r, Q)|TN (Q)|ψk (r, Q)i − ℏ
kt (Q) = 0
∂Q
∂Q
n
n
n
k6=j
(
Problem still not solvable.
(ii) The static or Longuet-Higgins Representation:
Htotal (r, Q)Ψt (r, Q) = Et Ψt (r, Q)
[Te (r) + U (r, Q) + TN (Q) + V (Q) − Et ]Ψt (r, Q) = 0
X
[Helec (r, Q0 ) + ΔU (r, Q) + TN (Q) + V (Q) − Et ]
ψk0 (r, Q0 )χSkt (Q) = 0.
k
By similar manipulations, realizing that
0
ψj (r, Q0 )|ψk0 (r, Q0 ) = δgk
2 ∂ 0
∂ 0
0
0
ψj (r, Q0 ) ψ (r, Q0 )
ψ (r, Q0 ) = ψj (r, Q0 ) ∂Qn k
∂Q2 k
n
=0
for all j, k
Page 5
5.80 Lecture # 3 3 Supplement
one obtains
TN (Q) + V (Q) + ǫ0j (Q0 ) + ψj0 (r, Q0 )|ΔU (r, Q)|ψj0 (r, Q0 ) − Et
X
×χSjt (Q) +
ψj0 (r, Q0 )|ΔU (r, Q)|ψk0 (r, Q0 ) χkt (Q) = 0
k6=j
Adiabatic Approximations
Definition: Adiabatic refers to any vibronic approximation scheme in which the wavefunction is factorized
in the form:
ΨAD (r, Q) = ψ(r, X)χAD (Q)
One must distinguish between several adiabatic schemes:
Any adiabatic scheme is valid only if the “effective potential surface” is well separated from all other
potential surfaces.
(i.e. concept of a potential surface for the nuclear motion has meaning only if the adiabatic separation is
a sufficiently good approximation for the description of the molecular state under consideration)
Three Commonly Encountered Adiabatic schemes
I. The Born-Huang (BH) Adiabatic Approximation
In the set of dynamical differential equations, eliminate the coupling terms between
D
D
Xjt
(Q) and Xkt
(Q) where k 6= j
by assuming
hψj (r, Q)|TN (Q)|ψk (r, Q)i = 0
∂ ψk (r, Q) = 0
ψj (r, Q) ∂Qn for k =
6 j
for all k, j.
Thus, the decoupled equations become
effective potential
z
}|
{
BH BH
[TN (Q) + V (Q) + ǫj (Q) + hψj (r, Q)|TN (Q)|ψj (r, Q)i]χBH
jt (Q) = Ejt χjt (Q)
which implies
BH
ΨBH
jt (r, Q) = ψj (r, Q)χjt (Q).
II. The Born-Oppenheimer (BO) Adiabatic Approximation
In the set of dynamical differential equations, eliminate the coupling terms between
D
χD
jt (Q) and χkt (Q) where k 6= j
Page 6
5.80 Lecture # 3 3 Supplement
by assuming
hψj (r, Q)|TN (Q)|ψk (r, Q)i = 0
∂ ψk (r, Q) = 0
ψj (r, Q) ∂Qn for all j, k
for all j, k.
Thus, the decoupled equations become
effective potential
z
}|
{
BO BO
{TN (Q) + V (Q) + ǫj (Q) }χBO
jt (Q) = Ejt χjt (Q)
which implies
BO
ΨBO
jt (r, Q) = ψj (r, Q)χjt (Q).
III. The Crude Adiabatic (CA) Approximation
In the set of static differential equations, eliminate the coupling terms between
χSjt (Q) and χSkt (Q) where j 6= k
by assuming
0
ψj (r, Q0 )|ΔU (r, Q)|ψk0 (r, Q0 ) = 0 for k =
6 j.
Thus, the decoupled equations become
CA CA
TN (Q) + V (Q) + ǫ0j (Q) + ψj0 (r, Q0 )|ΔU (r, Q)|ψj0 (r, Q0 ) χCA
jt (Q) = Ejt χjt (Q)
which implies
0
CA
ΨCA
jt (r, Q) = ψj (r, Q0 )χjt (Q).
Page 7
5.80 Lecture # 33 Supplement
Crude
Adiabatic Approximation
Born-Oppenheimer
Adiabatic Approximation
Born-Huang
Adiabatic Approximation
Adiabatic
Wavefunction
0
CA
ΨCA
jt (r, Q) = ψj (r, Q)χjt (Q)
BO
ΨBO
jt (r, Q) = ψj (r, Q)χjt (Q)
BH
ΨBH
jt (r, Q) = ψj (r, Q)χjt (Q)
Electronic
Equation
[Te (r) + U (r, Q0 )]ψj0 (r, Q0 )
= 0j (Q)ψj0 (r, Q0 )
[Te (r) + U (r, Q)]ψj (r, Q)
= j (Q)ψj (r, Q)
[Te (r) + U (r, Q)]ψj (r, Q)
= j (Q)ψj (r, Q)
Vibrational
Equation
[TN (Q) + V (Q) + j (Q) + ψj0 (r, Q0 )|ΔU (r, Q)|ψj0 (r, Q0 ) ]
CA CA
×χCA
jt (Q) = Ejt χjt (Q)
[TN (Q) + V (Q) + j (Q)]
BO BO
×χBO
jt (Q) = Ejt χjt (Q)
[TN (Q) + V (Q) + j (Q)
+ ψj (r, Q0 )|TN (Q)|ψj (r, Q)]
BH BH
×χBH
jt (Q) = Ejt χjt (Q)
ψj (r, Q)|TN (Q)|ψk (r, Q) = 0
and
ψj (r, Q) ∂Q∂ N ψk (r, Q) = 0
ψj (r, Q)|TN (Q)|ψk (r, Q) = 0 for k = j
and
ψj (r, Q) ∂Q∂ N ψk (r, Q) = 0
Approximations
Utilized
0
0
ψj (r, Q0 )|ΔU (r, Q)|ψk0 (r, Q0 ) = 0 for k = j
Page 8
5.80 Lecture # 3 3 Supplement
Example of Corrections within the Adiabatic Approximation
Improvement from the Crude Adiabatic (CA) Approximation to the Born-Oppenheimer (BO) Approximation
(Herzberg-Teller vibronic coupling)
ψ 0 (r, Q0 ) −→ ψ(r, Q)
The difference in the electronic Hamiltonians comes from the term ΔU (r, Q) where:
X ∂U (r, Q) 1 X ∂ 2 U (r, Q)
ΔU (r, Q) ≈
Qn +
Qn Qm + . . .
∂Qn
2 n,m ∂Qn ∂Qm 0
0
n
By perturbation theory
ψi (r, Q) ≈ ψi0 (r, Q0 ) +
X
Aji (Q)ψj0 (r, Q0 )
j6=i
where
0
ψj (r, Q0 )|ΔU (r, Q)|ψi0 (r, Q0 )
;
Aji (Q) =
ǫ0i (Q0 ) − ǫ0j (Q0 )
thus
BO
ΨBO
ir (r, Q) = ψi (r, Q)χir (Q)
h
i
X
≈ ψi0 (r, Q0 ) +
Aji (Q)ψj0 (r, Q0 ) χBO
ir (Q).
Corrections of Adiabatic Schemes to Non-Adiabatic Schemes
Goal: To express the total non-adiabatic wavefunctions in terms of adiabatic wavefunctions
via non-degenerate perturbation theory:
Ψir (r, Q) = ΨAD
ir (r, Q) +
X
ckt,ir ΨAD
kt (r, Q)
kt6=ir
where
AD
Ψkt (r, Q)|H′ (r, Q)|ΨAD
ir (r, Q)
ckt,ir =
.
AD − E AD
Eir
kt
The perturbation operator represents the breakdown of the adiabatic approximation:
H′ (r, Q) = Htotal (r, Q) − HAD (r, Q)
X
AD AD
ΨAD
= Htotal (r, Q) −
Ψkt (r, Q)
kt (r, Q) Ekt
kt
This leads to Born-Huang (BH) Coupling and Born-Oppenheimer (BO) Coupling.
e 1 A2 ← X
e 1A1 Absorption Transi­
The Transition Moment of the A
tion in Formaldehyde
The transition moment between adiabatic wavefunctions is given by
n
o
AD
AD
b
Mjt;ir
= ΨAD
jt (r, Q)|O(r)|Ψir (r, Q)
D
E
AD
b
ψ
(r,
Q)|
O(r)|ψ
(r,
Q)
= χAD
(Q)
χ
(Q)
.
j
i
jt
ir
Page 9
5.80 Lecture # 3 3 Supplement
To proceed, need to know Q-dependence of electronic integral.
e 1 A2 ← X
e 1 A1 transition
Let us apply this, with the Born-Oppenheimer, Adiabatic representation; to the A
of formaldehyde.
Lowest Singlet Electronic States in H2 CO
Energy (ev)
State Designation
State Number
1
0
3.50
7.08
7.97
9.45
A1
A2 (n, π ⋆ )
1
B2 (n, σ ⋆ )
1
A1 (π, π ⋆ )
1
B1 (σn, π ⋆ )
0
1
2
3
4
1
Assume that the electronic eigenfunction of the ground state can be expressed in terms of a non-mixed
crude adiabatic function:
BO
ΨBO
0t (r, Q) = ψ0 (r, Q)χ0t (Q)
≈ ψ00 (r, Q0 )χCA
0t (Q).
Perform a “Herzberg–Teller” expansion of the wavefunction for the first excited singlet state:
BO
ΨBO
1r (r, Q) = ψ1 (r, Q)χ1r (Q)
X
Aj1 (Q)ψj0 (r, Q0 )
ψ1 (r, Q) ≈ ψ10 (r, Q0 ) +
j>1
where
0
ψj (r, Q0 )|ΔU (r, Q)|ψ10 (r, Q0 )
Aj1 (Q) =
ǫ01 (Q0 ) − ǫ0j (Q0 )
and
ΔU (r, Q) ≈
X ∂U (r, Q) ∂Qn
n
Thus
Aj1 (Q) =
h
D
i E
∂U(r,Q) 0
0
ψ1 (r, Q0 )
X ψj (r, Q0 ) ∂Qn
0
ǫ01 (Q0 ) − ǫ0j (Q0 )
n
=
Qn .
0
X
Qn
γjn1 Qn .
n
Now
ψ1 (r, Q) ≈ ψ1◦ (r, Q0 ) +
XX
γjn1 Qn ψj◦ (r, Q0 )
j>1 n
and the Born-Oppenheimer Adiabatic wavefunction for the first excited singlet state becomes:
BO
ΨBO
1r (r, Q) = ψ1 (r, Q)χ1r (Q)


XX
n
≈ ψ10 (r, Q0 ) +
γj1
Qn ψj0 (r, Q0 ) χCA
1r (Q).
j>1 n
Page 10
5.80 Lecture # 3 3 Supplement
The transition moment now becomes:
n
o
BO
BO
b
MBO
=
Ψ
(r,
Q)|
O(r)|Ψ
(r,
Q)
0t;1r
0t
1r


D
E X X
0
b 0
n

= χCA
γj1
Qn ψj0 (r, Q0 ) χCA
1r (Q) .
0t (Q) ψ0 (r, Q0 ) O(r) ψ1 (r, Q0 ) +
j>1 n
Simplification yields:
D
E
0
0
CA
b
MBO
=
ψ
(r,
Q
)|
O(r)|ψ
(r,
Q
)
χCA
0
0
0t;1r
0
1
0t (Q)|χ1r (Q)
D
E
XX
0
CA
b
+
γjn1 ψ00 (r, Q0 )|O(r)|ψ
χCA
j (r, Q0 )
0t (Q)|Qn |χ1r (Q) .
j>1 n
Now consider the coefficients γjn1 :
γjn1 =
D
h
i E
0
ψj0 (r, Q0 ) ∂U(r,Q)
ψ1 (r, Q0 )
∂Qn
0
ǫ01 (Q0 ) − ǫ0j (Q0 )
.
Since the Hamiltonian must be invarient under all symmetry operations,
∂U (r, Q)
must transform as Qn .
∂Qn
0
Given that ψ10 (r, Q0 ) transforms as A2 , it is easy to find the appropriate combinations of Qn and ψj0 (r, Q0 )
such that γjn1 does not vanish via symmetry. Three non-zero coefficients are obtained:
4
γ21
,
5
γ41
,
6
γ41
.
Thus the transition moment for H2 CO can be written, more explicitly, as
Note that:
D
E
0
0
CA
b
MBO
χCA
0t;1r = ψ0 (r, Q0 )|O(r)|ψ1 (r, Q0 )
0t (Q)|χ1r (Q)
D
E
4
CA
b r)|ψ20 (r, Q0 ) χCA
+ γ21
ψ00 (r, Q0 )|O(
0t (Q)|Q4 |χ1r (Q)
D
E
CA
5
0
b
χCA
+ γ41
ψ00 (r, Q0 )|O(r)|ψ
0t (Q)|Q5 |χ1r (Q)
4 (r, Q0 )
D
E
0
CA
6
b
ψ00 (r, Q0 )|O(r)|ψ
(r,
Q
)
χCA
+ γ41
0
4
0t (Q)|Q6 |χ1r (Q)
D
E
1
CA
b
= 1 A1 (r, Q0 )|O(r)|
A2 (r, Q0 ) χCA
0t (Q)|χ1r (Q)
D
E
4
1
1
CA
b
+ γ21
A1 (r, Q0 )|O(r)|
B2 (r, Q0 ) χCA
0t (Q)|Q4 |χ1r (Q)
D
E
5
1
1
CA
b
+ γ41
A1 (r, Q0 )|O(r)|
B1 (r, Q0 ) χCA
0t (Q)|Q5 |χ1r (Q)
D
E
6
1
CA
b r)|1 B1 (r, Q0 ) χCA
+ γ41
A1 (r, Q0 )|O(
0t (Q)|Q6 |χ1r (Q) .
µz =⇒ A1
mz =⇒ A2
µx =⇒ B1
mx =⇒ B2
µy =⇒ B2
my =⇒ B1